Simplify and expand the following expression: $ \dfrac{5p}{p - 7}-\dfrac{4p + 7}{4p + 5} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(p - 7)(4p + 5)$ Multiply the first term by $\dfrac{4p + 5}{4p + 5}$ $ \begin{align*} \dfrac{5p}{p - 7} \times \dfrac{4p + 5}{4p + 5} & = \dfrac{(5p)(4p + 5)}{(p - 7)(4p + 5)} \\ & = \dfrac{20p^2 + 25p}{(p - 7)(4p + 5)}\end{align*} $ Multiply the second term by $\dfrac{p - 7}{p - 7}$ $ \begin{align*} \dfrac{4p + 7}{4p + 5} \times \dfrac{p - 7}{p - 7} & = \dfrac{(4p + 7)(p - 7)}{(4p + 5)(p - 7)} \\ & = \dfrac{4p^2 - 21p - 49}{(4p + 5)(p - 7)}\end{align*} $ Now we have: $ = \dfrac{20p^2 + 25p}{(p - 7)(4p + 5)} - \dfrac{4p^2 - 21p - 49}{(4p + 5)(p - 7)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{20p^2 + 25p - (4p^2 - 21p - 49)}{(p - 7)(4p + 5)} $ $ = \dfrac{20p^2 + 25p - 4p^2 + 21p + 49}{(p - 7)(4p + 5)} $ $ = \dfrac{16p^2 + 46p + 49}{(p - 7)(4p + 5)}$ Expand the denominator: $ = \dfrac{16p^2 + 46p + 49}{4p^2 - 23p - 35}$